The long periods caused by Jupiter, or Jupiter and Saturn in the motion of a Trojan are studied by numerical integration. Orbital inclinations between 0° and 50° are considered. Results on the length of the periods and on their main effects in the osculating elements are presented. It turns out that secular resonances can be important for the evolution of Trojan orbits.

Resonant asteroidal motion is investigated over 17 000 years in a three-dimensional elliptical model Sun-Jupiter-Asteroid averaged by Schubart's method à la Poincarè. Orbits remain trapped in the resonance over this period. The various stability mechanisms are discussed. With respect to their behaviour of ω and ῶ, our resulting orbits reveal 5 distinct classes. These 5 classes can be described in Schubart's topology for the planar problem. As compared to the planar case, eccentricities of orbits in the three-dimensional model vary more strongly. This is an important result for the problem of the delivery of meteorites.

The motion of the orbits of Trojan asteroids are investigated. Four asteroids around L5 are shown to have librating perihelion. A criterion for the libration of the perihelion is derived and expressed by the initial conditions. This criterion is solved for the initial conditions in two special cases. Thus regions in the configuration space of the initial conditions yielding libration of the orbits of Trojan asteroids are established.

The outcomes of asteroidal catastrophic collisions are strongly affected by the target asteroid's gravity, since only the fragments escaping with initial velocities higher than the target's escape velocity are not reaccumulated into “rubble pile” remnants. This idea can be compared with the observational evidence on the properties of family asteroids in several ways : (1) the shape and spin period of the “reaccumulated” family asteroids will roughly fit the relationships valid for self-gravitating fluid bodies; (2) the relative velocities of the few escaping fragments arising from a breakup event marginally overcoming self-gravity will often have an anisotropic distribution, affecting the final distribution of orbital elements; (3) the amount of mass which in a given family escaped to “infinity” will be correlated with the target's size, since only for objects larger than ~ 100 km self-gravity plays an important role. These predictions are discussed and compared with the available data.

In most applications resonance problems are implicitely or explicitely modelized by the Fundamental Model of Resonance i.e. the pendulum, characterized by its Hamiltonian function:

The ages of asteroid families are estimated by using a second order secular perturbation theory. Their ages are rather young. To explain these results we have some possible interpretations. The first interpretation is that in the theory of secular perturbation we do not include the effects of mutual perturbations of asteroids. The second is that we can regard these values as the minimum ages.

A linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

The continuation of symmetric periodic orbits can be described in terms of “symmetry functions”; the branching of the zero-level lines in a neighbourhood of a critical point gives rise to the transition from “first kind” to “second kind” periodic orbits. When the families are parametrized with the Jacobi integral, the bifurcations can be described as “catastrophes” of the generating functions. However bifurcations of higher order are more complex than the generic catastrophes with one parameter: both symmetric and asymmetric bifurcations occur.

In this way the symmetric periodic orbits that do not have close approaches to the secondary body can be described in an analytic way and their stability can be deduced from simple bifurcation rules. However numerical experiments are required to determine the “natural termination” of the families.

In the three-dimensional restricted three-body problem, the existence of resonant periodic solutions about L4 is shown and expansions for them are constructed for special values of the mass parameter, by means of a perturbation method. These solutions form a second family of periodic orbits bifurcating from the triangular equilibrium point. This bifurcation is the evolution, as μ varies continuously, of a regular vertical bifurcation point on the corresponding family of planar periodic solutions emanating from L4.

An algorithm for the numerical determination of asymmetric periodic solutions of the planar general three body problem is described. The elements of the “variational” matrix which are used in this algorithm are computed by numerical integration of the corresponding “variational” equations. These elements are also used in the study of the linear isoenergetic stability. A number of asymmetric periodic orbits are presented and their stability parameters are given.

Periodic orbits in a fixed frame are constructed in the vicinity of nonperiodic solutions of the non perturbed problem. In a first phase, approximate initial conditions are found and in a second phase more accurate initial conditions obtained are used in order to check the periodic orbit by numerical integration of the three-body problem. Some peculiar solutions are found, for example, orbit with nearly zero angular momentum. A study of stability of periodic solutions is proposed with an approximation of the monodromy matrix Φ (T,o),not requiring numerical integration of the 6x6 variational linear system. Finally, some numerical problems of period determination are outlined.

The anisotropic Kepler problem has a group of symmetries with three generators; they are symmetries respect to zero velocity curve and the two axes of motion’s plane. For a fixed negative energy level it has four homothetic orbits. We describe the symmetric periodic orbits near these homothetic orbits. Full details and proofs will appear elsewhere (Casasayas-Llibre).

The properties of the characteristic curves of several families of periodic orbits, in a conservative dynamical system of two degrees of freedom, symmetric with respect to both axes, are reviewed. The two main types of families are presented. One sees that the pattern of the characteristics in the exact resonance case is similar to that of the near resonance case except for the basic characteristic . The form of the characteristics can be found theoretically by means of the second integral.

The consequences for hierarchical stability of almost circular almost coplanar, low perturbation orbits in an n-body hierarchical dynamical system is discussed. It is shown that frequent close approaches to mirror conditions with subsequent reversing of perturbations is ensured by such properties. The part played by near commensurabilities in mean motion is also discussed, the Sun-Jupiter-Saturn case being taken as an example.

New results are obtained for the linear stability of the triangular points in the elliptic restricted problem using the Hill equations which describe the infinitesimal motion around L4,L5. Also the shape of the 4Π-periodic solutions along the transition curves in the μ-e plane is investigated .

By simple symmetry and change-of-scale considerations the topology of the level manifolds of the classical integrals of the N-body problem is shown to depend only on the value of the integral z = c2h (total angular momentum squared times total energy). For every hierarchical structure given to the N bodies the problem can be described as a set of N-l perturbed two-body problems by means of a fitted Jacobian coordinate system; in this setting the Easton inequality, relating potential, momentum of inertia and the z integral, is easily rederived. For N=3 the confinement conditions due to this inequality can be described, in a pulsating synodic reference system, as level lines of a modified potential function on a plane.

In the elliptic restricted three body problem an invariant relation between the velocity square of the third body and its potential is studied for long time intervals as well as for different values of the eccentricity. This relation, corresponding to the Jacobian integral in the circular problem, contains an integral expression which can be estimated if one assumes that the potential of the third body remains finite. Then upper and lower boundaries for the equipotential curves can be derived. For large eccentricities or long time intervals the upper boundary increases, while the lower decreases, which can be interpreted as shrinking respectively growing zero velocity curves around the primaries.

We study the cantorian structure of the successive intersections of the invariant manifolds of infinity (parabolic orbits) with a certain surface of section. The first of these intersections is computed numerically. The structure of the set of orbits of capture or escape after n binary collisions is given.

Outgoing asymptotic orbits at a collinear Equilibrium point of the elliptic restricted three-body problem are constructed analytically and numerically and those which intersect perpendicularly the axis of symmetry are selected by means of a numerical procedure of differential corrections. Due to the symmetry properties of the problem, the latter orbits terminate asymptotically back to the equilibrium point and therefore provide a kind of “asymptotic trapping” at the unstable equilibrium, a trapping mechanism based on "long duration of passage" rather than stability of motion.

We study the general version of the inverse problem for planar trajectories and for autonomous dynamical systems possessing three integrals, i.e., for a given three-parametric family of curves f(x,y,a,b)=c we find the potential V(x,y) for which these curves are orbits of a unit mass. All possible cases, depending on the preassigned function f, are classified and in each case the necessary and sufficient conditions for the.existence of a solution are established. Among the examples is the case of the Keplerian conic sections which is studied in detail.